Research papers

Current research interests

The Wholeness Axiom and Foundations of Mathematics. Dr. Corazza has published 15 journal articles concerning large cardinals and the attempt to provide a mathematical foundation for them. Large cardinals are extremely large infinite sets that sometimes arise in mathematical practice. Yet, the standard mathematical foundation cannot account for them—large cardinals are in principle underivable from the ZFC axioms. Corazza’s research begins from the observation that the assertion “there exists an infinite set” is an assertion about the existence of something “local” in the universe. He observes that a proper understanding of the mathematical infinite requires a deeper, “global” expression for the existence of the infinite. Lawvere in 1969 formulated such a global statement, showing that the existence of a certain type of functor j: V -> V is equivalent to the existence of an infinite set (where V is the universe of mathematics). Blass later showed that a somewhat stronger functor of the same type is equivalent to the existence of a measurable cardinal (one of the better known large cardinals). Corazza’s work then extends the properties of these functors to the fullest possible extent. His Wholeness Axiom asserts the existence of an elementary embedding j: V -> V with the additional property (essentially) that the restriction of j to any set is itself a set. ZFC+Wholeness Axiom is sufficient to derive all large cardinals. Inspiration for the Wholeness Axiom derives from Maharishi’s Vedic Science in which it is seen that all expressions in the universe arise from the self-interacting dynamics of wholeness, the field of consciousness itself. In a similar fashion, it can be shown that all sets in the universe are the expression of the dynamics of j: V->V, the wholeness embedding. Moreover, as in Maharishi discusses in his commentary on the Ved, individual expressions arise in the “collapse” of an unbounded value to its point value, and subsequent expansion from point to infinity. In the case of the Wholeness Axiom this is seen by the fact that j has a critical point—a first point moved—and this point gives rise to a compact expression (analogous to the Ved itself) from which all particular sets are seen to emerge.

Self-referral Foundation of Computation. In Maharishi’s Vedic Science, it is observed that all dynamics in the universe are the expression of self-referral dynamics of the pure field of existence. Nature’s computation is exact and without mistake because it is based on the dynamics of wholeness itself interacting with itself. Mathematically, in a parallel fashion, it can be shown that all computation can be proven to be represented in a recursive algorithm, and all recursive algorithms f are known to have a purely self-referral “definition” in the form F(f) = f, where F is a recursive operator from S to S, where S is the set of all partial functions from N to N. Moreover, these self-referral dynamics are given expression in the untyped lambda-calculus. In the lambda calculus, it can be shown that every term F has a fixed point f—i.e. a term for which Ff=f. Using this fact and the representability of arithmetic in the lambda calculus, one shows that this fixed point point property of terms in fact lies at the basis for all recursion in mathematics and computer science, and therefore at the basis of all computation. Dr. Corazza is in the process of writing a book for a general readership on these topics, and discusses these points in his course on Algorithms.

Design of Rules Engines. During seven of Dr. Corazza’s years in the software development industry, his work was centered around devleoping a good design for mid-scale rules engines. It has been recognized in recent years that a significant business asset for many modern companies is their business rules. In legacy systems, rules underlying business processes are handled in software implementation simply as if… then statements sprinkled through millions of lines of code. The need to represent rules in an external repository and in a language that lends itself to modification by the non-programmer have led to the emergence in recent years of rules languages. Dr. Corazza developed one such language in a recent project for an insurance company, and has more recently upgraded his approach by making use of the Java-certified Jess language. Dr. Corazza teaches a seminar on the Jess language and good rules engine design principles.

Work experience

Dr. Corazza has worked as a contractor for Google and e-Trade in Silicon Valley, and at several insurance companies in the United States, with 15 years experience as a Java engineer. Some projects in these roles include:

Development of a framework for a large-scale, high-volume website to support localization (e-Trade)

Development of a highly flexible custom rules engine, including a rules language, to support rules management for an enterprise-level customer-service application in the insurance industry (Mutual of Omaha)

Research

Dr. Corazza’s research interests are logic, set theory, category theory, large cardinals, sets of reals, and the lambda calculus. His research associated with teaching and the software industry are in the areas of algorithm analysis, rules engines, and software engineering.